Simplify ($\sqrt2 + \frac{1}{\sqrt3})(\sqrt2 - \frac{1}{\sqrt3}$)

Analysis

The expression given is $(\sqrt{2} + \frac{1}{\sqrt{3}})(\sqrt{2} - \frac{1}{\sqrt{3}})$. This is a classic algebraic problem that follows the Difference of Two Squares pattern.

Concept

The difference of two squares identity states that:

In this problem:

• $a = \sqrt{2}$
• $b = \frac{1}{\sqrt{3}}$

Step-by-Step Solution

Step 1: Apply the identity Using the formula $a^2 - b^2$:

Step 2: Evaluate the squares

• $(\sqrt{2})^2 = 2$
• $(\frac{1}{\sqrt{3}})^2 = \frac{1^2}{(\sqrt{3})^2} = \frac{1}{3}$

Step 3: Subtract the values Now, we have:

Step 4: Find a common denominator To subtract, express 2 as a fraction with a denominator of 3:

So,

Conclusion

The result of the simplification is $\frac{5}{3}$. Comparing this to the options provided, it matches option B.

Exam Tip

Whenever you see an expression in the form $(x + y)(x - y)$, don't waste time using the "FOIL" method (multiplying term by term). Simply square the first term, square the second term, and subtract them. It saves time and reduces the risk of calculation errors!